Global and Local Scaling Limits for Linear Eigenvalue Statistics of Jacobi $\beta$-Ensembles

TL;DR

This paper investigates the asymptotic behavior of linear eigenvalue statistics in Jacobi ensembles, revealing their connection to sine and Bessel kernels in different spectral regions, and establishing relations among various ensemble types.

Contribution

It provides a detailed analysis of the moment-generating functions for Jacobi ensembles, expressing them as Fredholm determinants and linking their limits to well-known kernels.

Findings

Mean and variance relate to sine kernel in the bulk spectrum

At the spectrum edge, they relate to Bessel kernel

Relations established between different Jacobi ensembles

Abstract

We study the moment-generating functions (MGF) for linear eigenvalue statistics of Jacobi unitary, symplectic and orthogonal ensembles. By expressing the MGF as Fredholm determinants of kernels of finite rank, we show that the mean and variance of the suitably scaled linear statistics in these Jacobi ensembles are related to the sine kernel in the bulk of the spectrum, whereas they are related to the Bessel kernel at the (hard) edge of the spectrum. The relation between the Jacobi symplectic/orthogonal ensemble (JSE/JOE) and the Jacobi unitary ensemble (JUE) is also established.